What About Transport of Utilities Between Space Colonies?

Much of what I write here is an extension of the idea that our conventional vision of orbital space colonies involves an impractically small population size. This is natural to think about in terms of movement of people, but it’s even more painful if you compare it to how we provide mundane utilities to cities on Earth today.

What Utilities Are Involved?

Space colonies may have somewhat different needs than Earth cities, but we're all human, after all. Provision of utilities is all very scale-dependent. In my reference size of an artificial gravity tube, there are about 20,000 people. Compare to a city on Earth, what utilities are distributed on a scale larger than this? The answer is “just about everything”.

Things that must be distributed between tubes:

• Electricity
• Communication
• Goods
• Industrial Fluids
• People
• Water / Sewage

I’m using a catch-all of “industrial fluids” to denote anything that is remotely similar to the role that oil plays in the world today. We transport oil through pipelines because there is SO MUCH of it to move that we need good efficiency. A space colony wouldn’t use hydrocarbons in the same way, but they may use fuels like Hydrogen. All these things would necessarily need to be managed among several tubes at once, and possibly throughout the entire gravity balloon. But I do want to make mention of at least one thing I can imagine for which the multi-tube assumption is not (can not be) true for.

Things produced and consumed inside a single tube:

• Light

The psychological impact of light can’t be discounted, but when we’re talking about a colony that has a wall over 10 km thick, then it’s pretty impractical to pipe natural sunlight through this barrier, and then pipe it through complicated snaking channels into a cluttered and rotating tube.

But Physically, Where Will These Go?

Artificial gravity tubes have an access limitation around the outside due to the flow dividers to reduce drag. That means that all utilities, like people, need to go in through the tube openings on either side. For most of these utilities, they will need to access some connection points near the axial line, and then be distributed to the inner surface through “vertical” pipes or elevators (elevators are in the case of batch processes).

The challenge is that you want to avoid moving things in batch processes as much as you possibly can. Batch processes are tremendously economic. A bath process going through an airlock would be much worse - thus, the entire motivation for a gravity balloon.

How Can They be Moved?

Electricity probably has the simplest answer, and is a total cop-out. We use slip rings all the time for electric machinery, some of which are the largest units on the grid, supplying more than the population of one of these tubes. The power constraint itself should not be a problem, but there will likely be a voltage constraint. Several kV shouldn’t be a problem, and this might constrain the transformer architecture a little bit. Let me elaborate a little more specifically. Imagine that there are 2 levels of transformers to get electricity from the zero-gravity high-voltage electric transmission system to someone’s home on the inside of the tube. You will need one transformer to step it down from, say, 750 kV outside the tube to, say 10 kV, going into the tube. After that, you’ll need another transformer to take it down to 100-200 V for residential use, as needed. This is because the slip-rings connect the tube electric distribution system to the outside by brushes that slide along a conductor while the tube spins. These brushes wear out, and they will wear out faster at higher voltages. There is a practical limit, and also other hazards due to the larger footprint of the slip rings.

Communication also has an easy cop-out, which is wireless technology. Again, this might be a slight headache for the experts (in this case, networking experts) who build the system. Alternatively, you could (again) use slip rings in this case as well. Even better, it might be possible to create a coupling for a fiber-optic cable that allows both ends to rotate relative to each other. This wouldn’t even necessarily have to connect near the tube’s axial line (I guess the same could be said for electricity, but it’s a harder sell).

People - I’ve mentioned moving people in and out in other posts. In short, you will need elevator transport to and from the axial line, although I am partial to the idea of entering the tube through a literal slide.

Goods - again, there’s no other choice but to move shipping containers through the axial line and lower/raise through elevators. Combined with people, the staging areas for these are sure to occupy the majority of the bottleneck of the tube-end openings.

Air - I have not mentioned among the others, because it must be handed in the same distribution system that temperature control operates in. It’s not a “visible” distribution system, although barriers to control the movement of air will be a major source of clutter in the tubes.

Industrial fluids - this is where the problem gets hard. If you did need to transport oil or natural gas, you would likely have to do it in a batch process along with the rest of the goods transported. There is still the possibility for a local distribution system on the inner surface of the tube with storage just for that 20,000 person (or however many) population. A rotating joint for this type of stuff is not easy.

Water / Sewage - You would think that the same principle might apply here - that there’s no other option but to ship in water trucks through the axial line and have them disembark, unload, and exit alongside sewage trucks taking the used water back out to shared process facilities among many tubes. But I think this is where it gets interesting.

Water is different than something like natural gas because contact with ambient air isn’t necessarily bad, and because it has a definite phase difference. This allows for a different kind of coupling… a weird one that I can’t imagine would ever come up except for in situations like the tubes in a gravity balloon. In short, I think you would use something like a toroidal water-garden at a partial gravity level. There, you can allow a glorified straw to add water to the partially-filled torus, or to suck up sewage.

At this point, it’s getting weird, and I like that. But the conversation about water distribution is going to be more involved. This is something I hope to cover in my next post, and I think it touches on some very cool and very novel concepts.

But Would the Nested Flow Dividers Really Work?

People coming from different backgrounds will have trouble with different parts of the gravity balloon concept for space colonies. However, the one criticism I am most excited to receive is that the flow dividers might not work. This came up on a reddit thread sharing this blog. I am overjoyed to hear this criticism - because it means that the critic has understood all the big details. They are up to speed, and that means they are ready for the meat of the conversation.

Now, having something spin in an atmosphere presents issues of its own, and the author proposes a complicated scheme involving nested shells to avoid turbulence. This feels like the sketchiest part of it to me - there's a lot of handwaving involved.

For too long, I have neglected to argue the core mechanical details on this blog. I can't give it the full treatment it deserves in limited time, but I'll break out the big guns (even if that only means labeling them).

Why the Balloon is Not Disputed

Other scientists and engineers have already covered the basic physical mechanism of a gravity balloon with no rotating structures inside. In every case, educated people who looked at the problem said "yeah, of course that would work". The core prediction comes from Newtonian gravity.

Take a moment to appreciate this fact: A gravity balloon construction has never existed. Even if someone tried to replicated it within a present-day space station, the other (mostly molecular) forces would dwarf self-gravitation. It is a purely hypothetical construction. Yet we are all agreed (all of the informed, for whatever it matters) with 100% certainty that it would work.

Conservative Approach to Flow Dividers

What is the "sketchy" part of the flow dividers? Like any engineering, the concept originates directly from the equations, given specific assumptions.

• Equations - Parallel plate turbulent flow (or laminar, if needed)
• Assumptions - The geometry and movement of the flow dividers

You probably need a fluids expert to comment on this. One problem might be that those equations are not exact... but this is unconvincing. Turbulent flow models don't run the risk of dramatically underestimating the drag. The transition point from laminar to turbulent is also highly uncertain. That would chip away at the laminar flow designs I have entertained before.

Also, there's more to flow than the global sheer forces. We have eddy currents. Those can form resonant patterns of certain kinds, you could posit that those might be destructive. But that claim is just plain wrong - because the exact problem has been studied before. It's called Taylor-Couette flow. For the most part, this leaves the flow circling in cylinders between the sheets. No, I don't have the exact flow description for the (very turbulent, very big) geometry described here, but there's nothing spectacular about the flow regime.

Geometry is the most challenging part of this all. The flow solution is all well-and-good, but it assumes that the sheets are in certain places. This requires them to be held there. That could be difficult, maybe even impossible. That might demand large steel scaffolding holding the flow dividers in place, along with mechanical joints and wheels to maintain separation between the nested sheets. This could become quite expensive. I'm not even willing to concede that this scenario makes it totally nonviable in all foreseeable circumstances.

Just take a moment to accept, however, that demanding assumptions of large structural supports (to resist air currents) is the most conservative academically honest position you could take. The flow regimes have already been in literature. All I'm asking is to apply them to a fictional geometry.

Very Liberal Approaches

The sell gets difficult when we start attempting to strip down those supports for maintaining the geometry. As I've argued, you can try using flimsy sheets. Perhaps you apply some positive pressure to them so that they hold a pseudo-rigid cylinder shape. But maybe not. We can just handwave these complications away.

In fact, there are two components to maintaining the geometry.

1. Keeping the flow dividers from colliding or jostling
2. Keeping the flow dividers shape in tact

Violating #2 might also imply a violation of #1, but I'm not worried as much about #2. Balancing the pressure in each layer will likely maintain shape as a side effect.

I've received one interesting response that seems to argue that the sheets may not even be necessary because the transition to laminar flow isn't clearly defined and may not necessarily exist if certain precautions are made (what exactly, I don't know). That sort of position is too liberal for me.

You might even build flow dividers with massive holes in them. Flow dividers which are more of a suggestion for the flow than a solid rule might be entirely sufficient. As for myself, I pull back a little bit from that vision. There is a lot of energy in the system, and the movement makes it difficult to identify a clear lines to the isobars in a mostly open system. If this were simply cylindrical geometry, I would be more inclined to the idea, but the end tapers wreck havoc on the flow complexity. I'm mainly speaking from intuition here, and I think that partially-open flow dividers are in the engineering battleground.


Mechanical Stability (the meat of the discussion)

There is something called the "wedge effect", but it might not be called exactly this depending on the source. Lots of large machinery levitate a rotor on a fluid. Some of that machinery rotates at tremendously high speeds. Essentially, the combination of the rotation in conjunction with the

You can find plenty of literature on the subject. Go look at chapters 2 and 3 of this book for some basic theory. This is what makes me largely an optimist, because it argues for a mostly passive answer to component #1. Additionally, it's an answer that is quite hopeful to aid in the concerns of component #2. Oscillations aren't really going to lead up to a Bernoulli effect like a naive reading might seem to suggest. The forces in involved have most to do with friction and a roughly static pressure profile.... if its fully laminar. As we get into turbulent territory, there is some wiggle room for pessimists, but it's more unknown than anything else.

Well, I just wanted to get that out there. Consider the surface to be scratched.

Construction of a Space Mega-City from a Bald Balloon Frame

Over on Orion's Arm, they put out a very cool article on gravity balloons (link also in sidebar nav now). In the process, they just happened to propose a new construction method as well. I had not gone this direction because I thought the asteroid in-situ use is critical for nearest-term viability because it avoids large material movement. That assumption, like all others, can change depending on unknown future capabilities. Not only is the method possible, it's actually more material  efficient due to a roughly 2/3^rds power scaling benefit due to decreasing the surface area to volume ratio. In other words, it lets you combine smaller asteroids into one colony which will be more mass-efficient than any of the individual asteroids alone.

Also, there is some amazing eye candy far beyond anything in this blog. The things which look like sesame seeds are broken pieces of asteroids. The balloon is a membrane with a small pressure tolerance, and those rocks are placed (or sprinkled) on evenly so that the pressure is increased through mutual gravitation. The red thing is a specific radiator design.

Image and copyright of Steve Bowers at Orion's Arm

Some specific constraints on how you can move the rock material will come up with this method. However, I see two competing frameworks for developing a model. One fundamental constraint would be to avoid any angles larger than the tumbling angle for rocks. Another constraint is that your material strength must be enough to support some curvature due to ballooning out between the rocks. You could use any Platonic solid to create a model of this, although I would prefer to just start with a tetrahedron. Also, I'm not sure if the balloon should be considered to have some linear mass, or the rock placement should form some kind of regular summation. There could be some similarities to the rockfill problem I wrote about, but with even more potential directions. I don't think this would be an easy problem, but that depends on the approach and how the math turns out.

If you consider the balance between the membrane's material strength and the number of trips needed to add the rocks, that forms a very coherent mathematical problem and I would love to have a crack at it sometime.

Impact of Rotating Asteroid on Orientation of Artificial Gravity Tubes

In a NASA Spaceflight forums discussion, I had a new complication pointed out to me that I'm surprised I had not thought to cover on this blog before. The artificial gravity tubes are rotating, but the asteroid is as well. We can distill the implications into 3 options. Either:

• the tube's axis of rotation will appear to move relative to the walls as the asteroid
• you will have to use some complicated form of "compound" rotation
• you will align the axis for rotation of all tubes parallel to the asteroid's axis of rotation

Aligning all tubes in the same direction is the most obvious and cheapest option. In the last post on global heat removal, I inadvertently designed this in. If you follow the implications a little bit, that implies that the majority of the heat exchangers are toward the asteroid's poles. This is intriguing because that would be the ideal place for space radiators in the first place. However, if the colony is pushing the spherical black-body limit, then it's not as good because the GEO radiator is toward the pole anyway.

Actually, an ability to precess was already designed into some O'Neil colony designs. With 2 artificial gravity tubes near to each other, they can push and pull so that they change the direction they face. This was proposed as a way to keep facing the sun, which obvious means they turn a full circle roughly every year (depends on the orbital position). For the asteroid rotation, we could be talking about rotation rates of 8 hours or less. Whether the forces involved and the unwanted acceleration on the surface are small enough is a question that is still unresolved.

This is a type of topic which I might return to sometime, but I've decide that I'm going to do a few micro-posts from the long list of things in my backlog where I introduce the subject in a few paragraphs.

Simplified Fully Turbulent Argument for Flow Dividers

Recently thinking about the core technology for a gravity balloon habitat (the friction buffers in my prior terminology), I realized that I have yet to confirm that I have convinced anyone of its function. Further deconstructing the disconnect, I'm not sure how many people have been convinced of the problem (a few, at least), and only that subset of people are eligible to understand the proposed solution in the first place. My current guess is that time is just needed to process it. This concept took me quite some digesting at first.

The underlying equations can be as simple or as complicated as you like. A true freely rotating cylinder isn't nice to work with, and for the laminar solution it even has a paradox for it. Since my prior post which handled the issue was relatively general over all flow regimes, it was also less easy to understand. Here, I'll simplify things to show the scaling that will most likely be relevant.

Extremely Simplified Argument

If you'll indulge only the most basic dimensional analysis intrinsic to almost all turbulent drag calculations, the motivation and operation behind the friction buffers becomes clear very quickly. Imagine a baseball flying through the air. Most physicists wouldn't hesitate to quickly categorize the force as being proportional to v^2. If you want to think in terms of units of power, it's a very quick step to say that (power) = (velocity) x (force). That means that power has a form of v^3 in the same way that we said that force has a form of v^2. With this, we can get a dependence on the number of sheets.

Variables Needed:

• N: number of sheets
• tau: shear pressure on outer wall of habitat (Pascals)
• f: Darcy friction factor
• rho: density of air
• v: velocity of air within a single channel
• V: outer velocity of the habitat itself
• P: power per unit area

The argument is accomplished here in its most simple form. Our equation for shear pressure (the retarding force per area on the cylinder) only involves the velocity out of all variables impacted by the number of sheets. Since the relevant channel velocity is divided by the number of stages, power ultimately goes with 1/N^2. Clearly, adding more layers of sheets will reduce friction.

Including Friction Factor

But why does this not involve the width of the channel? Because to a first approximation, the width of the channel doesn't matter, and that is hidden as a dependency of the friction factor "f". Of course this is a bad assumption. Intuitively the friction between two sheets moving parallel depends on the distance between them. The root of the prior formulation is that fluid velocity only increases logarithmically from the distance to the wall. Given high enough Reynold's number ln(x) is kind of a little bit flat-ish.

Clearly this is unsatisfactory, so to remedy this we merely rewrite things with a form for the friction factor - which is where this dependency lies. This is commonly illustrated in the Moody Chart. Here I have reproduced that chart from the Colebrook-White equation.

Many approximations exist for the friction factor, but we know almost exactly what regime we'll be in. I plotted those points for different numbers of sheets, along with an approximation for the reference tube design. We'll plug that approximation into the prior format.

As a minor detail, Moody's chart has the roughness in it as well. This will penalize adding more sheets technically, but it doesn't matter in the end. The reason is because decreasing the channel width increases the size of the roughness to channel size. However, even if we have 20 sheets, that leaves something like 5 meters. The sheet itself should only be a few milometers in my view, so clearly the roughness can't be more than this. Comparing to the values displayed here that's off the chart. Might as well consider it smooth.

New Variables:

• d: half-width of a single channel
• D: width of entire friction buffer region

With this specific approximation, we can plug back into the previous equation in order to have a very definite form for the shear stress. To go from this to power, multiply by the overall V. This involves a few steps and a little bit of math, but at the end we'll have a very useful form.

Let me stress that this is an accurate representation of our situation as long as we're in the turbulent regime. What is this constant on the top though? That's the power you would have in the case of N=1 for a specific reference case. That still doesn't represent a rotating cylinder in a free atmosphere, because it utilizes a standard channel width of about 100 meters.

I find it entirely arguable that you would have to install a flow arrester like this if you built any kind of in-atmosphere artificial gravity tube... because if you don't you'll have very high air currents to deal with. Using this flow arrester will increase drag somewhat from the scenario where you don't use it. Anyway, it forms a reference case that I need in order to make this math comprehensible.

So the basic proposition rehashed is:

The economic case seems pretty clear. Even in Manhattan people average something like 30 square meters of land per person. In the reference case, that would amount to 15 kW per person, which would likely dominate energy consumption. This doesn't mean it's impossible. You could do this, but if you did, thermal and energetic constraints would be setting your engineering limits at just about every turn. Clearly you wouldn't go this route if a better alternative existed - and it does. The formula is quite simply to scale the number of flow divider sheets to achieve the necessary power level.

Power Calc Stage-by-Stage

One might also be confused by the calculation of power. Indeed, for the fluid dynamics calculations we used "v", lower case v, which is the half-velocity within a channel. But to get total power consumption, we multiply shear stress by "V", the total velocity shift over the entire friction buffer region. This is justified because the torque to keep the tube rotating is applied relative to the stationary reference frame, but it is dragged on by the fluid forces present in only one layer.

But I don't expect everyone to believe that right away, so I'll show the other approach as well. Let the index _i mean that a quantity is for a single stage. To sum up power, we have to add a factor of 2 because these aerodynamic forces exist on both sides of the sheet.

Variables:

• P_i : power for a single stage
• P : total power summed for all stages
• tau : shear stress, there is only one value of this, because force is transmitted over all layers

I don't think this is particularly profound, but perhaps it needed to be stated.

Appendix

The expression for "tau" uses 4 f in it. This is done in order to employ the Darcy friction factor. If you used the Fanning friction factor instead, you would just use "f".

Reynolds number contains "d" for the width of the channel. We know there are N channels, however, the equation for shear stress doesn't apply directly to Couette flow, but instead a channel with one free end. This is mathematically the same as half of one of our channels, so the operative d would be D/2. However, hydraulic diameter in Reynold's number must be found by using 4Ax/Dx, where Ax and Dx are the flow area and diameter. This comes out to 2 times the width of the channel in question. Combining these two factors, we see that the linear dimension to be used in Reynold's number is D/N.

Global Air Heat Transport in a Gravity Balloon

The complete scheme for heat transport within a gravity balloon habitat with artificial gravity cylinders can be segmented by following general outline. This tells the story of the path of heat as it's generated where people live through the point when it eventually gets emitted out into space.

1. Heat generation occurs inside a artificial gravity tube
2. Hot air flows out of the outlet window of the tube
3. Hot air flows to the walls of the gravity balloon
4. Heat energy passed through a heat exchanger to some non-air medium
5. Non-air medium passes through a space radiator as the ultimate heat sink

This post will focus on how the air currents transfer the heat from the outlet of a tube to the wall of the gravity balloon where the heat exchanger then picks up, #3 in the above list. The previous post outlined one valid scheme for #2. In the past I've entertained several bad ideas for a scheme that accomplishes #3. Notably, since the currents need to occur over such a large area and only need to move so slowly, I imagined an impossibly large fan many kilometers across. If possible, apparently absurd or extraordinary schemes should be rejected (unless the problem it solves is, itself, extraordinary). In this case, just like in the last part, a much more elegant solution presents itself.

Outer Sheet as the Air Movement Mechanism

Just like heat transport out of the artificial gravity tube, it would be preferable to reduce the number of parts, so if there's a device already called for by the design we would prefer to configure things so that one device solves multiple problems at the same time. Thankfully, that exact thing is possible. We will pump the air ultimately by using the driving force applied to keep the artificial gravity tubes spinning.

But first, I must specify that I imagine a stationary lattice which is connected to the asteroid rock and can accept forces. This is absolutely necessary for the motors to push against which keep the cylinders spinning, but they shouldn't be particularly difficult to build. A key distinction now comes in how we configure the outermost layer of the friction reducing flow dividers (friction buffers). That outermost layer could conceivably be connected to the stationary lattice structure. Up until this point my math has assumed this is the case, but that was only done for mathematical simplicity.

In fact, it would be best to allow the outermost layer to freely rotate. For some numbers, let's say that the speed of the habitat on the inner surface of the tubes is 100 miles per hour and there are 20 flow dividers. Each stage then sees roughly 5 mph relative speed to the next stage. What will be the velocity of the outermost layer? Answer: considerably more than 5 mph.

To understand this, I will offer a concept of "resistance to movement" between each layer of flow dividers. Given that the stages have constant spacing between them, this is roughly the same for them all (with some difference due to varying radii). However, the distance between the outermost layer and the bulk atmosphere isn't something which can be clearly defined. If we imagine the point of "r=infinity" to be another flow divider, the spacing between that and the outermost flow divider is clearly more than the spacing between the other dividers.

Given that the outermost sheet sees less resistance to its motion, its natural preference will be to couple more strongly to the speed of the habitat (100 mph) than to the bulk atmosphere (0 mph) than would be predicted by it's share of the speed divided evenly among the flow dividers (5 mph). But we have yet another pesky effect that we have to deal with. If multiple tubes are in the same general vicinity and their air currents compliment each other, this could increase the ultimate speed that the outermost layer equilibrates to. Those are potentially two factors which push it above the 5 mph prediction for this case.

For all of these reasons, I believe that the outermost layer will have some kind of "brakes" on it which prevents it from speeding up too much. Even better- if the motor is attached to the outermost layer this will improve efficiency somewhat, although it would require an additional motor to keep that layer rotating slightly relative to the stationary lattice. Even if you simply threw away the extra energy, the scheme will see a very small efficiency reduction and will work just fine. Accept a little extra energy consumption or a little extra complexity - the choice is yours.

All of this is only to say that the velocity of the outer layer can be selected to some degree.

Direction of Thermal Gradient

The ability to move flow in the local vicinity between tube would be pointless anyway if there wasn't some coherent path that takes the tube's hot air exahust to the wall's heat exchanger in order to ultimately dissipate the heat. Because of that, I'm making the obvious claim that every tube has its flow connected to two different channels. It obtains its intake air from one channel and exhausts hotter air into the other channel.

Picturing this takes a little bit of creativity, and my illustration skills might be lacking. The divider between the global channels (different from the friction buffers themselves) is a 2D sheet that cuts accross multiple artificial gravity tubes. For the most part, this plane cuts through the tube's axis of rotation. We just apply a slight skew or deviation in order to allow the exhaust and intake ends to connect to their respective channels. Here is my illustration of the situation:

Some attributes are exaggerated in this figure. The exhaust and intake windows are only 10% of the habitat's radius in most reference designs I've used. Thus, this slant we're working with might only be slight. Alternatively, the sheet could be completely parallel to the axis of rotation, and the windows will connect into small local depressions. Well "small" in this case would still be around 25 meters, but that's small compared to other stuff involved.

Hot Channel Calculations

Moving on, we need to figure out what limitations this scheme places on the overal gravity balloon size and/or population. I will return to my reference design in order to illustrate a pattern of air flow. Flow dividers must be added between a series of artifical gravity tubes, and then air flows in different directions on both sides. This produces an interlocking pattern of air currents.

Putting that in perspective of the entire balloon, I have the following image in mind. Here, I have included the presumed heat exchangers on the wall. I hope that makes it clear what kind of back-and-fourth pattern the flow is traveling in, and how it gets hottest right before it gets to the heat exchanger.

We can set the temperatures to whatever is desired at the start and endpoint for these flows. However, since the flow travels in a straight line, it seems fairly clear that the highest temperature change will be experienced by the line of colonies that goes straight through the center of the sphere. It is this row that sets our limit.

I would imagine this limit will be around 10 degrees C or Kelvin. Perhaps 20 degrees. If you refer to the heat transport within the tubes, that is certain to be on the order of 5 degrees, and some colony will experience the extremes of these temperatures. As such, it's probably best to keep it to 10.

As long as we're accepting my reference design, that has 22,000 people, and I'll stick to the claim that they're using 2 W each, for a total of 44 MW. On average, there is one colony per 1 km^3 lattice. The colony takes up a small fraction of the total volume (about 27%). Thus, the effective flow area is about 0.86 km^2. With all these ingredients, we can formulate the heat balance relationship for the balloon-level flows.

Heat Balance for Global Heat Transport Channels

Selecting the Delta_T value is related to biological and comfort limits. Heat transport within the tubes themselves is already known to require about 5 Kelvin of temperature change in order to employ natural circulation at the desired population density levels. If we add much more variation, then we could have undesirable large temperature swings. I would imagine that a number under 5 Kelvin would be acceptable.

As I previously argued, v is a tunable variable up to a certain limit, and that limit relates to the degree of friction reduction in maintaining spin. Presumably, this would be under about 3 m/s, but it could be a good deal more. With this piece of info, we have a fairly strong argument for what the bounds on these variables are.

For a given Qdot, we can find the number of colonies which can be served. Referring to the previous illustrations of the scheme, there is on average one colony per linear kilometer of flow chanel. That means that Qdot/(44 MW) will yield the maximum number of colonies it can serve. Since the hot channel's length is equal to the diameter of the gravity balloon, dividing by 2 can give us the radius of the maximum size (in km of radius) that the scheme can thermally support. I present those cases in this table:

Thermal Limitation of Gravity Balloon

Based on Global Heat Removal Channels

These are large sizes, and the assumptions about Delta_T and v are quite conservative.

Implications

This problem, in particular, would seem to be extremely easy to solve and pose little constraints on engineering of other related systems. There are multiple parameters that you could scale up in order to efficiently globally circulate air in a gravity balloon for just about any practical scale.

A large area for the flow channel was key in making this so easily solvable. Even if the generous parameters for maximum balloon size were not sufficient for someone's desires, there are multiple ways of pushing the envelope further. For instance, increasing the spacing between artificial gravity tubes or reducing the number of friction buffers. Even these methods would only need to be applied in a select regions which are subject to the extremes of the hot channel temperatures.

Another major benefit of this system is that the driving force is applied constantly. It is somewhat concerning that the flow path isn't completely straight, but I doubt that any really good solutions to this problem exist. You could conceive of extreme solutions, like flow divider sheets that are partially friction buffers for the tubes and partially friction buffers for the global flow channels, but this is certainly not necessary. The most important benefit of the continuous driving force as well as frictional losses is that no major pressure differential exists within the whole of the gravity balloon. This means that no hardened airtight doors will be necessary for people and goods passing between the different flow channels, which is sure to happen often.

Natural Circulation Heat Removal from Artificial Gravity Tubes

In a previous post I concluded that natural circulation of air was an attractive method of removing heat produced by a population living within an artifical gravity tube within a gravity balloon. This notion was still very vague, so I want to place some numbers on that, and also potentially define the scale at which it would be economical.
To summarize the idea, the airflow goes in at one end and out the other with no pumping. This is made possible by the fact that heat is produced by the inhabitants in their everyday lives, and also by the designed geometry of the cylinder which lets the hot air rise up to the outlet while preventing cold air coming from the inlet from reaching the center.

Forms loss (I'll sometimes call k-loss) is a means of grouping together various resistances to flow along a flow path. This is always referenced to a specific cross-section on the path, and I will reference it to the pinched open end. For a free jet condition, it is somewhere around 1.0 generally. In our case, we have a large stagnant atmosphere outside of the tube as well as a mostly stagnant atmosphere inside the tube. That causes both the inlet and outlet to be something close to free jet conditions... with a lot of qualifiers. Since the flow is expanding radially, it must also exchange a great deal of angular momentum with structures attached to the rotating tube, similar to the case of a centrifugal pump. This should substantially affect the k-loss value, but probably not by more than, say, a factor of 2. Given that we have 2 free jet conditions, I would most likely expect k to fall somewhere in the neighborhood of 2 to 4, but this is a highly imprecise science at this point. Thankfully, as long as it's somewhere close to that range it shouldn't critically wound our overall conclusions.

Tube radius, heat production, temperature range, end opening size, and air flow velocity are all important things which have very practical relevance to the design of an artificial gravity tube. Armed with some educated guesses for the k-loss factor, we can set constraints on these parameters. Firstly, I'll divide up these values which are absolute, fungible, and independent variables.

Parameters for air:

• density              rho0 ~ 1.3 kg/m3
• heat capacity    Cp ~ 1,005 J/(kg-K)

Relatively fixed variables

• temperature of the environment                    T0 ~ 293 K
• gravity in the living areas of the environment g = 9.8 m/s^2
• Power consumption per inhabitant    gamma ~ 2 kW
• Window edge radius relative to habitat radius  Rw/R ~ 0.1

Independent design variables

• Radius of the tube
• Change in temperature across the tubes
• Velocity of the air at the end seals
• Population of the society

Equations to Relate Variables

By definition, the k-loss equation is the following. This quantity represents the frictional pressure head fighting against the direction of flow.

Pressure Drop due to Friction

The essential idea of natural circulation is that heavy cold air flows down from the inlet to the surface habitat, and then less-dense warmed air rises from the habitat toward the center-line point. To find the change in density we must return to basic PV=nRT gas law concepts. Compared to the magnitude of the temperature change, the pressure changes very little relative to its environment value. Thus, to deal with the density change we can just imagine that it changes linearly with the temperature change.

Density Change Given Temperature Change

It is this density change which gives rise to the natural circulation driving force. This works by the analog of (Delta_P=rho g h) in constant Earth gravity. But gravity varies with radius in the case of artificial gravity. Since the driving force is the difference in hydrostatic pressure change with altitude, it goes with the change in density as opposed to absolute density.

Natural Circulation Driving Pressure

Driving force then exactly matches the frictional losses experienced over the flow path. Thus, we can set the two expressions to be equal. This constitutes the momentum balance for the natural circulation heat removal system.

Momentum Balance Final Form

Variables involved in design:

• Delta_T
• R
• v

With this relationship nailed down, we can consider the limitation on heat production. Along with this we have a litany other other supplementary relationships introduced. The mass flow rate through the tube is connected to the end window size. The area of the end window is related to the window's aspect ratio as well as the overall radius. Total heat production goes with total population as well as the per-capita energy intensity of the society (I call gamma).

Writing these all out and then combining them:

Population / Heat Relationship

Additional variables involved in design:

• P

So while we added another equation, we also added another free variable. In other words, this doesn't add any dimensionality to the problem, it's just an auxiliary equation that I'll use to calculate a population limitation given the other parameters.

Numerical Values

With more-or-less 3 variables and 1 equation, we have two degrees of freedom. The relationship is pretty straightforward but it's not very helpful in that form without comparing it to some reference designs or tangible speeds and sizes.

As a simple applcation of the equations, here are some values for 3 cases of radius, 3 cases of temperature, and 2 scenarios for the k-loss value. That is 3x3x2=18 total numbers. In each of those cases, we have dependent variables of "v" (the velocity at the end windows) and "P", the population.

For some further notes, I've included the velocity of the edge of the window for all the cases for different radii. These are assuming that the windows are 10% of the radius of the habitat surface. I've distinguished between that as "V edge" and the flow relevant to heat removal as "V flow". As you can see, the window edge velocities tend to be even higher than the outward and inward flow for the other parameters I've selected.  That, itself, might be a problem but it's a geometric consequence of the window size. The window could be made smaller while accepting some other sacrifices.

The population limit reported here is then divided by the livable area within a habitat. It is assumed (as in the reference design) that the length of the cylinder is equal to its diameter and no credit is taken for the are on the pinched ends.

I put NYC and Manhattan on this as well for a reference. Note that in my reference design for the artificial gravity tube, density is still incredibly high - about that of Manhattan. This applies for the scenario of Delta_T=5k, R=250m, and k=4, where the population constraint comes out to be about 20,000 people.

To me, this still seems to be about the most reasonable reference design. I will elaborate on that a bit more in the conclusion.

Carbon Dioxide Removal and Other Undesirables

The design principle of the gravity balloon is more-or-less to locate industrial facilities that don't need gravity (or strong gravity) within the open air microgravity environment between colonies. It is crucial that we can show that critical services (like heat removal) can be viably provided outside the gravity tubes. For heat removal, not only can this be done, but it can be done at incredibly low cost using natural circulation. But that's not all we have to worry about.

Possibly the most vital metric to control within a space habitat is carbon dioxide levels since this will cause negative health effects before lack of oxygen, however the limitation relative to the habitat's heat removal is less clear. Let's just look at the comparitative limits between these two. Consideration of the specifics of an artificial gravity tube isn't necessary. I'll just consider what temperature rise would also correspond to a dangerous rise in CO2 levels.

Certain specifiers are needed, but I'll consider the most active society possible in order to be conservative. A human doing normal work will emit 0.08 to 0.13 m3/h of CO2. Using this information as well as the scenarios I've defined, we can find the increase in CO2 parts per million (ppm) as the air flows from the inlet window to the outlet window. Here are my estimations:

Increase in CO2 Concentration

for a Given Rise in Temperature

• 2 K : 31 ppm
• 5 K : 78 ppm
• 10 K : 155 ppm

None of these are particularly deadly. Humans can easily tolerate increases this much or greater. However, these were only formulated based on the assumpting that people were consuming 2 kW on average. That was supposed to be a conservative assumption, but in this case lower values might put us in a bit of a bind. If that was reduced to a value closer to the biological limit of around 200 W instead, then the above temperature changes would correspond to a dramatically higher CO2 concentration rise. As such, it's plausable to create scenarios where CO2 removal would be the overriding constraint on the allowable population of the tube... but this probably wouldn't be likely under normal conditions.

Big Picture Conclusion

My pessimism in the last post on this subject is lessened substantially. We can state a number of relatively attractive combinations of parameters which would be economically desirable and physical plausible. However, there is still a bit of a tight design envelope to fit.

The heat production limit would likely constitute the gravity balloon's version of a "fire code". You could certainly pack more people into the tube, but the temperature would rise slowly. Except for some possibly extreme circumstances, it seems unlikely that CO2 removal would become more restrictive than heat removal.

I find it hard to argue against natural circulation as a means of cooling the tubes themselves. The benefits compared to the alternatives seem immense. The air flow rates are unlikely to surpass the speeds which will be encountered near the windows anyway, and being a fairly localized thing, I don't expect the end seals to have a dramatic impact on the overall drag anyway.

In fact, in some cases the air flow would be so low that in the center you couldn't rely on it to move out of the tube (starting at centerline) in a timely manner. For these cases, you would need a conventional transport system or elevator-like system. Since the heat production rate will vary throughout the day, this seems inevitable anyway.